A CHARACTERIZATION OF CERTAIN SHIMURA CURVES IN THE MODULI STACK OF ABELIAN VARIETIES. Eckart Viehweg & Kang Zuo. Abstract

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1 j. differential geometry 66 (2004) A CHARACTERIZATION OF CERTAIN SHIMURA CURVES IN THE MODULI STACK OF ABELIAN VARIETIES Eckart Viehweg & Kang Zuo Abstract Let f : X Y be a semistable family of complex abelian varieties over a curve Y of genus g(y ), and smooth over the complement of s points. If l denotes the rank of the nonflat (1, 0) part F 1,0 of the corresponding variation of Hodge structures, the Arakelov inequality says that 2 deg(f 1,0 ) l (2g(Y ) 2+s). The family reaches this bound if and only if the Higgs field of the variation of Hodge structures is an isomorphism. The latter is reflected in the existence of special Hodge cycles in the general fibre, and the base of such a family is a Shimura curve. In particular, for s 0, such a family must be isogenous to the l-fold product of a modular family of elliptic curves, and a constant abelian variety. For s = 0, if the flat part of the variation of Hodge structures is defined over the rational numbers, one finds the family to be isogenous to the product of several copies of a family h : Z Y, and a constant abelian variety. In this case, h : Z Y is obtained from the corestriction of a quaternion algebra A, defined over a totally real numberfield F, and ramified over all infinite places but one. In case the flat part is not defined over the rational numbers, one still can classify the corresponding variations of Hodge structures. This work has been supported by the DFG-Schwerpunktprogramm Globale Methoden in der Komplexen Geometrie. The second named author was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. CUHK 4239/01P). Received 06/15/

2 234 E. VIEHWEG & K. ZUO 0. Introduction Throughout this article, Y will denote a nonsingular complex projective curve, U an open dense subset, and X 0 U a smooth family of abelian varieties. We choose a projective nonsingular compactification X of X 0 such that the family extends to a morphism f : X Y, which we call again a family of abelian varieties although some of the fibres are singular. We write S = Y \ U, and = f 1 (S). Consider the weight 1 variation of Hodge structures given by f : X 0 U, i.e., R 1 f Z X0. We will always assume that the monodromy of R 1 f Z X0 around all points in S is unipotent. The Deligne extension of (R 1 f Z X0 ) O U to Y carries a Hodge filtration. Taking the graded sheaf one obtains the Higgs bundle (E,θ)=(E 1,0 E 0,1,θ 1,0 ) with E 1,0 = f Ω 1 X/Y (log ) and E0,1 = R 1 f O X. The Higgs field θ 1,0 is given by the edge morphisms of the tautological sequence f Ω 1 X/Y (log ) R1 f O X Ω 1 Y (log S) 0 f Ω 1 Y (log S) Ω 1 X(log ) Ω 1 X/Y (log ) 0. By [14] E can be decomposed as a direct sum F N of Higgs bundles with E 1,0 F ample and with N flat, hence for F i,j = E i,j F and N i,j = E i,j N the Higgs bundle E decomposes in (0.0.1) (F = F 1,0 F 0,1,θ 1,0 F 1,0) and (N 1,0 N 0,1, 0). For g 0 = rank(f 1,0 ) the Arakelov inequalities ([7], generalized in [20], [12]) say that (0.0.2) 2 deg(f 1,0 ) g 0 deg(ω 1 Y (log S)). In this note we will try to understand f : X Y, for which (0.0.2) is an equality, or as we will say, of families reaching the Arakelov bound. By Proposition 1.2, this property is equivalent to the maximality of the Higgs field for F, saying that θ F1,0 : F 1,0 F 0,1 Ω 1 Y (log S) isan isomorphism. As it will turn out, the base of a family of abelian varieties reaching the Arakelov bound is a Shimura curve, and the maximality of the Higgs field is reflected in the existence of special Hodge cycles on the general fibre. Before formulating a general result, let us consider two examples.

3 A CHARACTERIZATION OF SHIMURA CURVES 235 For families of elliptic curves, the maximality of the Higgs field just says that the family is modular (see Section 2). Proposition 0.1. Let f : E Y be a semistable family of elliptic curves, smooth over U Y.IfE Y is nonisotrivial and reaching the Arakelov bound, E Y is modular, i.e., U is the quotient of the upper half plane H by a subgroup of Sl 2 (Z) of finite index, and the morphism U C = H/Sl 2 (Z) is given by the j-invariant of the fibres. For S the only families of abelian varieties reaching the Arakelov bound are built up from modular families of elliptic curves. Theorem 0.2. Let f : X Y be a family of abelian varieties smooth over U, and such that the local monodromies around s S are unipotent. If S, and if f : X Y reaches the Arakelov bound, then there exists an étale covering π : Y Y such that f : X = X Y Y Y is isogenous over Y to a product B E Y Y E, where B is abelian variety defined over C of dimension g g 0, and where h : E Y is a family of semistable elliptic curves reaching the Arakelov bound. Results parallel to 0.2 have been obtained in [30] for families of K3- surfaces, and the methods and results of [30] have been a motivation to study the case of abelian varieties. As we will see in Section 4, Theorem 0.2 follows from the existence of too many endomorphisms of the general fibre of f : X Y, which in turn implies the existence of too many cycles on the general fibre of X Y X. We give an elementary proof of Theorem 0.2 in Section 4, although it is nothing but a first example for the relation between the maximality of Higgs fields, and the moduli of abelian varieties with a given special Mumford Tate group Hg, constructed in [17] and [18] (see Section 2). Proposition 0.3. Let f : X Y be a family of g-dimensional abelian varieties reaching the Arakelov bound. Assume that g = g 0,or more generally that the largest unitary local subsystem U 1 of R 1 f C X0 is defined over Q. Then there exists a finite cover Y Y,étale over U, and a Q-algebraic subgroup Hg Sp(2g, R), such that pullback family f : X = X Y Y Y is a semistable compactification of the universal family of polarized abelian varieties with special Mumford Tate group contained in Hg, and with a suitable level structure.

4 236 E. VIEHWEG & K. ZUO As a preparation for the proof of Proposition 0.3 we will show in Section 1, using Simpson s correspondence between Higgs bundles and local systems, that the maximality of the Higgs field enforces a presentation of the local systems R 1 f C X0 and End(R 1 f C X0 ) using direct sums and tensor products of one weight one complex variation of Hodge structures L of rank two and several unitary local systems. Proposition 0.3 relates families reaching the Arakelov bound to totally geodesic subvarieties of the moduli space of abelian varieties, as considered by Moonen in [16], or to the totally geodesic holomorphic embeddings, studied by Abdulali in [1] (see Remark 2.5 b). As in [21] one could use the classification of Shimura varieties due to Satake [22] to obtain a complete list of those families, and to characterize them in terms of properties of their variation of Hodge structures. We choose a different approach, less relying on the theory of Shimura varieties, and more adapted to handle the remaining families of abelian varieties (see Remark 2.5 c), as well as some other families of varieties of Kodaira dimension zero (see [33]). We first show that the decompositions of R 1 f C X0 and End(R 1 f C X0 ) mentioned above are defined over Q R. In case S it is then easy to see, that the unitary parts of the decompositions trivialize, after replacing Y by a finite étale cover Y (see 4.4). For S =, let us assume first that the assumptions made in 0.3 hold true. By [9] (see Proposition 6.3) they imply that the family is rigid, i.e., that the morphism from Y to the moduli stack of polarized abelian varieties has no nontrivial deformation, except those obtained by deforming a constant abelian subvariety. Mumford gave in [18] countably many moduli functors of abelian fourfolds, where Hg is obtained via the corestriction of an quaternion algebra, defined over a totally real cubic number field F. Generalizing his construction one considers quaternion division algebras A defined over any totally real number field F, which are ramified at all infinite places except one. Choose an embedding D = Cor F/Q A M(2 m, Q), with m minimal. As we will see in Section 5 writing d =[F : Q] one finds m = d or m = d +1. By 5.9 and 5.10 we get the following types of moduli functors of abelian varieties with special Mumford Tate group Hg = {x D ; xx =1}

5 A CHARACTERIZATION OF SHIMURA CURVES 237 and with a suitable level structure, which are represented by a smooth family Z A Y A over a compact Shimura curve Y A. Since we did not fix the level structure, Y A is not uniquely determined by A. So it rather stands for a whole class of possible base curves, two of which have a common finite étale covering. Example 0.4. Let Z η denote the generic fibre of Z A Y A. Then one of the following holds true: i. 1 <m= d odd. In this case dim(z η )=2 d 1 and End(Z η ) Z Q = Q. ii. m = d + 1. Then dim(z η )=2 d and: a. For d odd, End(Z η ) Z Q a totally indefinite quaternion algebra over Q. b. For d even, End(Z η ) Z Q a totally definite quaternion algebra over Q. Let us call the family Z A Y A a family of Mumford type. For d = 1 or 2 the examples in 0.4 include the only two Shimura curves of PEL-type, parameterizing: Moduli schemes of false elliptic curves, i.e., polarized abelian surfaces B with End(B) Z Q a totally indefinite quaternion algebra over Q (see also [25]). Moduli schemes of abelian fourfolds B with End(B) Z Q a totally definite quaternion algebra over Q. We will see in Section 6 that for g = g 0, up to powers and isogenies, the families of Mumford type are the only smooth families of abelian varieties over curves reaching the Arakelov bound. Theorem 0.5. Let f : X Y be a smooth family of abelian varieties. If the largest unitary local subsystem U 1 of R 1 f C X is defined over Q and if f : X Y reaches the Arakelov bound, then there exist: a. a quaternion division algebra A, defined over a totally real number field F, and ramified at all infinite places except one, b. an étale covering π : Y Y, c. a family of Mumford type h : Z = Z A Y = Y A, as in Example 0.4, and an abelian variety B such that f : X = X Y Y Y is isogenous to B Z Y Y Z Y. Things get more complicated if one drops the condition on the unitary local subsystem U 1 of R 1 f C X. For one quaternion algebra A, there

6 238 E. VIEHWEG & K. ZUO exist several nonisogenous families. Hence it will no longer be true, that up to a constant factor f : X Y is isogenous to the product of one particular family. Example 0.6 (See 5.11). Let A be a quaternion algebra defined over a totally real number field F, ramified at all infinite places but one, and let L be a subfield of F. Let β 1,...,β δ : L Q denote the different embeddings of L. For µ =[F : L] + 1 (or may be µ =[F : Q] in case that L = Q and µ odd) there exists an embedding Cor F/L A M(2 µ,l). As is well-known (see Section 5) for some Shimura curve Y such an embedding gives rise to a representation of π 1 (Y, ) inm(2 µ,l), hence to a local L system V L. Moreover there exists an irreducible Q-local system X Q = X A,L;Q for which X Q Q is a direct sum of the local systems V L L,βν Q. There exist nonisotrivial families h : Z Y with a geometrically simple generic fibre, such that R 1 h Q is a direct sum of ι copies of X Q. Such examples, for g = 4 and 8 have been considered in [9]. Here F is a quadratic extension of Q, L = F and ι = 1 or 2. For g = 8, i.e., ι =2, this gives the lowest-dimensional example of a nonrigid family of abelian varieties without a trivial subfamily [9]. A complete classification of such families is given in [21]. Theorem 0.7. Let f : X Y be a smooth family of abelian varieties. If f : X Y reaches the Arakelov bound, then there exist an étale covering π : Y Y, a quaternion algebra A, defined over a totally real number field F and ramified at all of the infinite places except one, an abelian variety B, and l families h i : Z i Y of abelian varieties with geometrically simple generic fibre, such that: i. f : X = X Y Y Y is isogenous to B Z 1 Y Y Z l Y. ii. For each i {1,...,l} there exists a subfield L i of F such that the local system R 1 h i Q Zi is a direct sum of copies of the irreducible Q-local system X A,Li ;Q defined in Example 0.6. iii. For each i {1,...,l} the following conditions are equivalent: a. L i = Q. b. h i : Z i Y is a family of Mumford type, as defined in Example 0.4.

7 A CHARACTERIZATION OF SHIMURA CURVES 239 c. End(X A,Li ;Q) 0,0 = End(X A,Li ;Q). Moreover, if one of those conditions holds true, R 1 h i Q Zi is irreducible, hence R 1 h i Q Zi = X A,Li ;Q. Here, contrary to 0.5, we do not claim that a component h i : Z i Y is uniquely determined up to isogeny by X A,Li ;Q and by the rank of R 1 h i Q Zi. We do not know for which g there are families of Jacobians among the families of abelian varieties considered in 0.2, 0.5 or 0.7, i.e., whether one can find a family ϕ : Z Y of curves of genus g such that f : J(Z/Y ) Y reaches the Arakelov bound. For Y = P 1 the Arakelov inequality (0.0.2) implies that #S 4. Our hope, that a family of abelian varieties with #S = 4 can not be a family of Jacobians, broke down when we found an example of a family of genus 2 curves over the modular curve X(3) in [13] (see also [37]), whose Jacobian is isogenous to the product of a fixed elliptic curve B with the modular curve E(3) X(3) (see Section 7). As mentioned already, this article owes a lot to the recent work of the second named author with Xiao-Tao Sun and Sheng-Li Tan. We thank Ernst Kani for explaining his beautiful construction in [13], and for sharing his view about higher genus analogs of families of curves with splitting Jacobians. It is also a pleasure to thank Ben Moonen, Hélène Esnault and Frans Oort for their interest and help, Ngaiming Mok, for explaining us differential geometric properties of base spaces of families and for pointing out Mumford s construction in [18], Bruno Kahn and Claus Scheiderer for their help to understand quaternion algebras and their corestriction, and Martin Möller and the referee for pointing out several misprints and ambiguities. This note grew out of discussions started when the first named author visited the Institute of Mathematical Science and the Department of Mathematics at the Chinese University of Hong Kong. His contributions to the final version (in particular to the proof of Theorems 0.5 and 0.7) were written during his visit to the I.H.E.S., Bures sur Yvette. He would like to thank the members of those three Institutes for their hospitality. 1. Splitting of C-local systems We will frequently use C. Simpson s correspondence between polystable Higgs bundles of degree zero and representations of the fundamental group π 1 (U, ).

8 240 E. VIEHWEG & K. ZUO Theorem 1.1 (C. Simpson [27]). There exists a natural equivalence between the category of direct sums of stable filtered regular Higgs bundles of degree zero, and of direct sums of stable filtered local systems of degree zero. We will not recall the definition of a filtered regular Higgs bundle ([27], p. 717), just remark that for a Higgs bundle corresponding to a local system V with unipotent monodromy around the points in S the filtration is trivial, and automatically deg(v) = 0. By([27], p. 720) the latter also holds true for local systems V which are polarisable C-variations of Hodge structures. For example, 1.1 implies that the splitting of Higgs bundles (0.0.1) corresponds to a decomposition over C (R 1 f Z X0 ) C = V U 1 where V corresponds to the Higgs bundle (F = F 1,0 F 0,1,τ = θ F0,1 ) and U 1 to (N = N 1,0 N 0,1,θ N = θ N = 0). Let Θ(N,h) denote the curvature of the Hodge metric h on E 1,0 E 0,1 restricted to N, then by [11], chapter II we have Θ(N,h N )= θ N θ N θ N θ N =0. This means that h N is a flat metric. Hence, U 1 is a unitary local system. In general, if U is a local system, whose Higgs bundle is a direct sum of stable Higgs bundles of degree zero and with a trivial Higgs field, then U is unitary. As a typical application of Simpson s correspondence one obtains the polystability of the components of certain Higgs bundles. We just formulate it in the weight one case. Recall that F 1,0 is polystable, if there exists a decomposition F 1,0 i A i with A i stable, and deg A i ranka i = deg F 1,0 rankf 1,0. Proposition 1.2. Let V be a direct sum of stable filtered local systems of degree zero with Higgs bundle (F = F 1,0 F 0,1,τ). Assume that τ F 0,1 =0, that τ 1,0 = τ F 1,0 : F 1,0 F 0,1 Ω 1 Y (log S) F Ω 1 Y (log S),

9 A CHARACTERIZATION OF SHIMURA CURVES 241 and that for g 0 = rank(f 1,0 ) (1.2.1) 2 deg(f 1,0 )=g 0 deg(ω 1 Y (log S)). Then τ 1,0 is an isomorphism, and the sheaf F 1,0 is polystable. Proof of 1.2. Let A F 1,0 be a subsheaf, and let B Ω 1 Y (log S) be its image under τ 1,0. Then A B is a Higgs subbundle of F 1,0 F 0,1, and applying 1.1 one finds deg(a) + deg(b) 0. Hence deg(a) = deg(b) + rank(b) deg(ω 1 Y (log S)) deg(b) + rank(a) deg(ω 1 Y (log S)) deg(a) + rank(a) deg(ω 1 Y (log S)). The equality (1.2.1) implies that deg(a) rank(a) 1 2 deg(ω1 Y (log S)) = deg(f 1,0 ), g 0 and F 1,0 is semistable. If deg(a) rank(a) = deg(f 1,0 ), g 0 rank(a) = rank(b) and deg(b) = deg(a). The Higgs bundle (F 1,0 F 0,1,τ) splits by 1.1 as a direct sum of stable Higgs bundles of degree zero. Hence (A B,τ A B ) is a direct factor of (F 1,0 F 0,1,τ). In particular, A is a direct factor of F 1,0. For A = F 1,0 one also obtains that τ 1,0 is injective and by (1.2.1) it must be an isomorphism. q.e.d. The local system R 1 f Q X0 on U = Y \ S is a Q-variation of Hodge structures with unipotent local monodromies around s S, obviously having a Z-form. By Deligne s semisimplicity theorem [4] it decomposes as a direct sum of irreducible polarisable Q-variation of Hodge structures V iq. More generally, if V is a polarized C-variation of Hodge structures, and V = V i, i a decomposition with V i an irreducible C-local system, then by [7] each V i again is a polarisable C-variation of Hodge structures. In both cases, taking the grading of the Hodge filtration, one obtains a decomposition of the Higgs bundle (E,θ)=(F 1,0 F 0,1,τ) (N 1,0 N 0,1, 0)

10 242 E. VIEHWEG & K. ZUO as a direct sum of sub-higgs bundles, as stated in 1.1. Obviously, each of the V iq again reaches the Arakelov bound. Our next constructions will not require the local system to be defined over Q. So by abuse of notations, we will make the following assumptions: Assumption 1.3. For a number field L C consider a polarized L-variation of Hodge structures X L of weight one over U = Y \ S with unipotent local monodromies around s S. Assume that the local system X = X L L C has a decomposition X = V U 1, with U 1 unitary, corresponding to the decomposition (E,θ)=(F, τ 1,0 ) (N,0)=(F 1,0 F 0,1,τ 1,0 ) (N 1,0 N 0,1, 0) of Higgs fields. Assume that V (or (F, τ)) has a maximal Higgs field, i.e., that τ 1,0 = τ F 1,0 : F 1,0 F 0,1 Ω 1 Y (log S) is an isomorphism. Obviously, for g 0 = rank(f 1,0 ) this is equivalent to the equality (1.2.1). Hence we will also say, that X (or (E,θ)) reaches the Arakelov bound. Proposition 1.4. If deg Ω 1 Y (log S) is even there exists a tensor product decomposition of variations of Hodge structures V L C T, with: a. L is a rank-2 local system. For some invertible sheaf L, with L 2 = Ω 1 Y (log S) the Higgs bundle corresponding to L is (L L 1,τ ), with τ L 1 =0and τ L given by an isomorphism τ 1,0 : L L 1 Ω 1 Y (log S). L has bidegree 1, 0, and L 1 has bidegree 0, 1. b. If g 0 is odd, L g 0 = det(f 1,0 ) and L is uniquely determined. c. For g 0 even, there exists some invertible sheaf N of order two in Pic 0 (Y ) with L g 0 = det(f 1,0 ) N. d. T is a unitary local system and a variation of Hodge structures of pure bidegree 0, 0. If(T, 0) denotes the corresponding Higgs field, then T = F 1,0 L 1 = F 0,1 L. In Section 6 we will need a slightly stronger statement.

11 A CHARACTERIZATION OF SHIMURA CURVES 243 Addendum 1.5. If in 1.4, there exists a presentation V = T 1 C V 1 with T 1 unitary and a variation of Hodge structures of pure bidegree 0, 0, then there exists a unitary local system T 2 with T = T 1 C T 2. In fact, write (T 1, 0) and (F 1,0 1 F 0,1 1,τ 1 ) for Higgs fields corresponding to T 1 and V 1, respectively. Then deg(t 1 )=0and 2 deg(f 1,0 1 ) rank(t )=2 deg(f 1,0 ) = g 0 deg(ω 1 Y (log S)) = rank(f 1,0 1 ) rank(t ) deg(ω 1 Y (log S)). So (F 1,0 1 F 0,1 1,τ 1 ) again satisfies the assumptions made in 1.4. Proof of 1.4. Taking the determinant of τ 1,0 : F 1,0 one obtains an isomorphism det τ 1,0 : det F 1,0 F 0,1 Ω 1 Y (log S), det F 0,1 Ω 1 Y (log S) g0, By assumption there exists an invertible sheaf L with L 2 =Ω 1 Y (log S). Since F 1,0 F 0,1, (det F 1,0 ) 2 Ω 1 Y (log S) g0 = L 2 g 0, and det F 1,0 L g 0 = N is of order two in Pic 0 (Y ). If g 0 is even, L is uniquely determined up to the tensor product with two torsion points in Pic 0 (Y ). If g 0 is odd, one replaces L by L N and obtains det F 1,0 = L g 0. By 1.2 the sheaf T = F 1,0 L 1 is polystable of degree zero. 1.1 implies that the Higgs bundle (T, 0) corresponds to a local system T, necessarily unitary. Choose L to be the local system corresponding to the Higgs bundle (L L 1,τ ), with τ 1,0 : L L 1 Ω 1 Y (log S). The isomorphism τ 1,0 : T L= F 1,0 induces an isomorphism φ : T L 1 F 0,1 Ω 1 Y (log S) F 0,1, F 0,1 L 2 such that τ 1,0 = φ (id T τ 1,0). Hence the Higgs bundles (F 1,0 F 0,1,τ) and (T (L L 1 ), id T τ ) are isomorphic, and V T C L. q.e.d.

12 244 E. VIEHWEG & K. ZUO Remark 1.6. i. If deg Ω 1 Y (log S) is odd, hence S, and if the genus of Y is not zero, one may replace Y by an étale covering, in order to be able to apply 1.4. Doing so one may also assume that the invertible sheaf N in 1.4 c), is trivial. ii. For Y = P 1 and for X reaching the Arakelov bound, #S is always even. This, together with the decomposition 1.4, for U = C g 0, can easily obtained in the following way: by 1.2, F 1,0 must be the direct sum of invertible sheaves L i, all of the same degree, say ν. Since τ 1,0 is an isomorphism, the image τ 1,0 (L i )iso P 1(2 s+ν) Ω P 1(log S). Since F 0,1 is dual to F 1,0 one obtains ν =2 s + ν, and writing L 1 i = τ 1,0 (L i ), ( (F 1,0 F 0,1,τ) O P 1(ν) O P 1( ν), ) τ. i i Consider now the local system of endomorphism End(V) ofv, which is a polarized weight zero L-variation of Hodge structures. The Higgs bundle (F 1,0 F 0,1,τ) for V induces the Higgs bundle (F 1, 1 F 0,0 F 1,1,τ 1, 1 τ 0,0 ) corresponding to End(V) =V C V, by choosing F 1, 1 = F 1,0 F 0,1, F 0,0 = F 1,0 F 1,0 F 0,1 F 0,1 and F 1,1 = F 0,1 F 1,0. The Higgs field is given by τ 1, 1 =( id) τ 1,0 τ 1,0 id and τ 0,0 = τ 1,0 id ( id) τ 1,0. Lemma 1.7. Assume as in 1.3 that X reaches the Arakelov bound or equivalently that the Higgs field of V is maximal. Let Fu 0,0 := Ker(τ 0,0 ) and Fm 0,0 = Im(τ 1, 1 ). Then there is a splitting of the Higgs bundle (F 1, 1 F 0,0 F 1,1,τ 1, 1 τ 0,0 ) =(F 1, 1 Fm 0,0 F 1,1,τ 1, 1 τ 0,0 ) (Fu 0,0, 0), which corresponds to a splitting of the local system over C End(V) =W U.

13 A CHARACTERIZATION OF SHIMURA CURVES 245 U is unitary of rank g0 2 and a variation of Hodge structures concentrated in bidegree 0, 0, whereas W is a C-variation of Hodge structures of weight zero and rank 3g0 2. τ 1, 1 : F 1, 1 Fm 0,0 Ω 1 Y (log S) and τ 0,0 : Fm 0,0 F 1,1 Ω 1 Y (log S) are both isomorphisms. Proof. By definition, (Fu 0,0, 0) is a sub-higgs bundle of (F 1, 1 F 0,0 F 1,1,τ 1, 1 τ 0,0 ). We have an exact sequence 0 Fu 0,0 F 0,0 τ 0,0 F 1,1 Ω 1 Y (log S). Since τ 1,0 id is surjective, τ 0,0 is surjective, and deg(fu 0,0 ) = deg(f 0,0 ) deg(f 1,1 ) rank(f 1,1 ) deg(ω 1 Y (log S)). By the Arakelov equality, deg(f 1,1 )=g 0 deg(f 0,1 )+g 0 deg(f 1,0 ) = g0 2 deg(ω 1 Y (log S)) = rank(f 1,1 ) deg(ω 1 Y (log S)) and one finds deg(fu 0,0 ) = deg(f 0,0 )=0. By 1.1 (Fu 0,0, 0), as a Higgs subbundle of degree zero with trivial Higgs field, corresponds to a unitary local subsystem U of End(V). The exact sequence 0 Fu 0,0 F 0,0 F 1,1 Ω 1 Y (log S) 0 splits, and one obtains a direct sum decomposition of Higgs bundles (F 1, 1 F 0,0 F 1,1,τ)=(F 1, 1 Fm 0,0 F 1,1,τ) (Fu 0,0, 0), which induces the splitting on End(V) with the desired properties. q.e.d. In 1.7 the local subsystem W of End(V) has a maximal Higgs field in the following sense: Definition 1.8. weight k, and let Let W be a C-variation of Hodge structures of (F, τ) = p+q=k F p,q, τ p,q

14 246 E. VIEHWEG & K. ZUO be the corresponding Higgs bundle. Recall that the width is defined as width(w) = Max{ p q ; F p,q 0}. i. W (or (F, τ)) has a generically maximal Higgs field if width(w) > 0 and if: a. F p,k p 0 for all p with 2p k width(w). b. τ p,k p : F p,k p F p 1,k p+1 Ω 1 Y (log S) is generically an isomorphism for all p with 2p k width(w) and 2p 2 k width(w). ii. W (or (F, τ)) has a maximal Higgs field if the τ p,k p in i) b. are all isomorphisms. In particular, a variation of Hodge structures with a maximal Higgs field can not be unitary. Properties 1.9. a. If W is a C-variation of Hodge structures with a (generically) maximal Higgs field, and if W W is a direct factor, then width(w ) = width(w) and W has again a (generically) maximal Higgs field. b. Let L and T be two variations of Hodge structures with L T of weight 1 and width 1, and with a (generically) maximal Higgs field. Then, choosing the bidegrees for L and T in an appropriate way, either L is a variation of Hodge structures concentrated in degree 0, 0, and T is a variation of Hodge structures of weight one and width one with a (generically) maximal Higgs field, or vice versa. Proof. For a) consider the Higgs field ( F p,q,τ p,q) ofw, which is a direct factor of the one for W. Since the τ p,q are (generically) isomorphisms, a) is obvious. In b) denote the components of the Higgs fields of L and T by L p 1,q 1 and T p 2,q 2, respectively. Shifting the bigrading one may assume that p 1 = 0 and p 2 = 0 are the smallest numbers with L p 1,q1 0 and T p 2,q2 0 and moreover that the corresponding q i 0. Since q 1 + q 2 = 1, one of q i must be zero, let us say the first one. Then T p 2,q 2 can only be nonzero, for (p 2,q 2 )=(0, 1) or = (1, 0) and L is concentrated in degree 0, 0. Obviously this forces the Higgs field of L to be zero. Then the Higgs field of L T is the tensor product of the Higgs field T 1,0 T 0,1 Ω 1 Y (log S)

15 A CHARACTERIZATION OF SHIMURA CURVES 247 with the identity on L 0,0, hence the first one has to be (generically) an isomorphism. q.e.d. Remark The splitting in 1.7 can also be described by the tensor product decomposition V = T C L in 1.4 with T unitary and L a rank two variation of Hodge structures of weight one and with a maximal Higgs field. For any local system M one has a natural decomposition End(M) =End 0 (M) C, where C acts on M by multiplication. Applying 1.7 to L instead of V, gives exactly the decomposition End(L) =End 0 (L) C. One obtains End(V) =T C T C L C L =(End 0 (T) C) C (End 0 (L) C) = End 0 (T) C End(T) C End 0 (L). Here End 0 (T) C is unitary and W = End(T) C End 0 (L) has again a maximal Higgs field. Remark If one replaces End(V) by the isomorphic local system V C V, one obtains the same decomposition. However, it is more natural to shift the weights by two, and to consider this as a variation of Hodge structures of weight 2. A statement similar to 1.7 holds true for 2 (V). Here the Higgs bundle is given by F 2,0 = F 1,0 F 1,0, F 1,1 = F 1,0 F 0,1 and F 0,2 = F 0,1 F 0,1. 2. Shimura curves and the special Mumford Tate group Lemma 2.1. Let L be a real variation of Hodge structures of weight 1 dimension 2, with a nontrivial Higgs field. Let γ L : π 1 (U, ) Sl(2, R) be the corresponding representation and let Γ L denote the image of γ L. Assume that the local monodromies around the points s S are unipotent. Then the Higgs field of L is maximal if and only if U = Y \ S H/Γ L. Proof. Writing L for the (1, 0) part, we have an nontrivial map (2.1.1) τ 1,0 : L L 1 Ω 1 Y (log S).

16 248 E. VIEHWEG & K. ZUO Since L is ample, Ω 1 Y (log S) is ample, hence the universal covering Ũ of U = Y \ S is the upper half plane H. Let Ũ = H ϕ H be the period map. The tangent sheaf of the period domain H is given by the sheaf of homomorphisms from the (1, 0) part to the (0, 1) part of the variation of Hodge structures. Therefore τ 1,0 is an isomorphism if and only if ϕ is a local diffeomorphism. Note that by Schmid [23] the Hodge metric on the Higgs bundle corresponding to L has logarithmic growth at S and bounded curvature. By the remarks following [28], Propositions 9.8 and 9.1, τ 1,0 is an isomorphism if and only if ϕ : Ũ H is a covering map, hence an isomorphism. Obviously the latter holds true in case Y \ S H/Γ L. Assume that ϕ is an isomorphism. Since ϕ is an equivariant with respect to the π 1 (U, )-action on Ũ and the Pρ L(π 1 (U, ))-action on H, the homomorphism ρ LZ : π 1 (U, ) Pρ LZ (π 1 (U, )) PSl 2 (R) must be injective, hence an isomorphism. So ϕ descend to an isomorphism ϕ : Y \ S H/Γ L. q.e.d. Proof of Proposition 0.1. h : E Y be the semistable family of elliptic curves, reaching the Arakelov bound, smooth over U. Hence L Z = R 1 h Z E0 is a Z-variation of Hodge structures of weight one and of rank two. Writing L for the (1, 0) part, we have an isomorphism (2.1.2) τ 1,0 : L L 1 Ω 1 Y (log S). Since L is ample, Ω 1 Y (log S) is ample, hence the universal covering of U is the upper half plane H. One obtains a commutative diagram ϕ H H ψ ψ j U C where j is given by the j-invariant of the fibres of E 0 U, where ψ is the quotient map H H/Sl 2 (Z), and where ϕ is the period map. 2.1 implies that ϕ : U H/ρ LZ (π 1 (U, ))

17 A CHARACTERIZATION OF SHIMURA CURVES 249 is an isomorphism, hence ρ LZ (π 1 (U, )) Sl 2 (Z) is of finite index, and E Y is a semistable model of a modular curve. q.e.d. Let us recall the description of wedge products of tensor products (see [10], p. 80). We will write λ = {λ 1,...,λ ν } for the partition of g 0 as g 0 = λ λ ν. The partition λ defines a Young diagram and the Schur functor S λ. Assuming as in 1.4 that L is a local system of rank 2, and T a local system of rank g 0, both with trivial determinant, one has k (L T) = S λ (L) S λ (T) where the sum is taken over all partitions λ of k with at most 2 rows and at most g 0 columns, and where λ is the partition conjugate to λ. Similarly, S k (L T) = S λ (L) S λ (T) where the sum is taken over all partitions λ of k with at most 2 rows. The only possible λ are of the form {k a, a}, for a k 2. By [10], 6.9 on p. 79, { S{k 2a} (L) =S S {k a,a} (L) = k 2a (L) det(l) a if 2a <k S {a,a} (L) = det(l) a if 2a = k. For k = g 0 one obtains: Lemma 2.2. Assume that det(l) =C and det(t) =C. a. If g 0 is odd, then for some partitions λ c, g 0 (L T) = g c=0 S 2c+1 (L) S λ2c (T). In particular, for c = g one obtains g 0 S g 0 (L) (T) =S g 0 (L) as a direct factor. b. If g 0 is even, then for some partitions λ c, g 0 (L T) =S g 0 (L) S {2,...,2}(T) g 02 1 c=1 S 2c (L) S λ2c (T). Lemma 2.3. Assume that L and T are variations of Hodge structures, with L of weight one, width one and with a maximal Higgs field, and with T pure of bidegree 0, 0.

18 250 E. VIEHWEG & K. ZUO a. If k is odd, H 0 (Y, ) k (L T) =0. b. If k is even, say k =2c, then for some λ c ) k H (Y, 0 (L T) = H 0 (Y,det(L) c S λc (T)). c. For k =2one has in b) λ 1 = {2}, hence S λ1 (T) = 2 (T). d. H 0 (Y,S 2 (L T)) = H 0 (Y,det(L) 2 (T)). Proof. S l (L) has a maximal Higgs field for l>0, whereas for all partitions λ the variation of Hodge structures S λ (T) is again pure of bidegree 0, 0. By 1.9 a), S l (L) S λ (T) has no global sections. Hence k (L T) can only have global sections for k even. In this case, the global sections lie in det(l) c S λc (T), for some partition λ c, and one obtains a) and b). For k = 2 one finds λ 1 = {2}. Ford) one just has the two partitions {1, 1} and {2}. Again, the direct factor S 2 (L) S 2 (T), corresponding to the first one, has no global section. q.e.d. Let us shortly recall Mumford s definition of the Hodge group, or as one writes today, the special Mumford Tate group (see [17], [18], and also [5] and [24]). Let B be an abelian variety and H 1 (B,Q) and Q the polarization on V. The special Mumford Tate group Hg(B) is defined in [17] as the smallest Q-algebraic subgroup of Sp(H 1 (B,R),Q), which contains the complex structure. Equivalently Hg(B) is the largest Q- algebraic subgroup of Sp(H 1 (B,Q),Q), which leaves all Hodge cycles of B B invariant, hence all elements η H 2p (B B,Q) p,p = [ 2p (H 1 (B,Q) H 1 (B,Q))] p,p. For a smooth family of abelian varieties f : X 0 U with B = f 1 (y) for some y U, and for the corresponding Q-variation of polarized Hodge structures R 1 f Q X0 consider Hodge cycles η on B which remain Hodge cycles under parallel transform. One defines the special Mumford Tate group Hg(R 1 f Q X0 ) as the largest Q-subgroup of Sp(H 1 (B,Q),Q) which leaves all those Hodge cycles invariant ([5], 7, or [24], 2.2).

19 A CHARACTERIZATION OF SHIMURA CURVES 251 Lemma 2.4. a. For all y U the special Mumford Tate group Hg(f 1 (y)) is a subgroup of Hg(R 1 f Q X0 ). For all y in the complement U of the union of countably many proper closed subsets it coincides with Hg(R 1 f Q X0 ). b. Let G Mon denote the smallest reductive Q-subgroup of Sp(H 1 (B,R),Q), containing the image Γ of the monodromy representation γ : π 0 (U) Sp(H 1 (B,R),Q). Then the connected component G Mon 0 of one in G Mon is a subgroup of Hg(R 1 f Q X0 ). c. If f : X Y reaches the Arakelov bound, and if R 1 f C X has no unitary part, then G Mon 0 =Hg(R 1 f Q X0 ). Proof. The first statement of a) has been verified in [24], 2.3., and the second in [16] 1.2. As explained in [5], 7, or [24], 2.4, the Mumford Tate group contains a subgroup of Γ of finite index, hence b) holds true. It is easy to see, that the same holds true for the special Mumford Tate group (called Hodge group in [17]) by using the same argument. Since the special Mumford Tate group of an abelian variety is reductive, a) implies that Hg(R 1 f Q X0 ) is reductive. So G Mon 0 Hg(R 1 f Q X0 ) is an inclusion of reductive groups. The proof of 3.1, (c), in [6] carries over to show that both groups are equal, if they leave the same tensors [ 2p (H ] η 1 (B,Q) H 1 (B,Q)) invariant. Let η H k (B,Q) be invariant under Γ, and let η be the corresponding global section of k (R 1 f Q X0 )= k (L T). By 2.3 a) and b), one can only have global sections for k =2c, and those lie in det(l) c S λc (T). In particular they are of pure bidegree c, c. The same argument holds true, if one replaces B and f : X Y by any product, which implies c). q.e.d.

20 252 E. VIEHWEG & K. ZUO For the Hodge group Hg(R 1 f Q X )=Hg Sp(2g, Q), as in Lemma 2.4 Mumford considers the moduli functor M(Hg) of isomorphy classes of polarized abelian varieties with special Mumford Tate group equal to a subgroup of Hg. He shows that M(Hg) admits a quasiprojective coarse moduli space M(Hg), which lies in the coarse moduli space of polarized abelian varieties A g. By Mumford ([17], Section 3, [18], Sections 1 2) M(Hg) = Γ\Hg(R)/K where K is a maximal compact subgroup of Hg(R), and Γ an arithmetic subgroup of Hg(Q). The embedding M(Hg) A g is a totally geodesic embedding, and M(Hg) is a Shimura variety of Hodge Type Hg. Let f : X 0 U be a family of abelian varieties with the special Mumford Tate group Hg(R 1 f Q X0 )=Hg. By Lemma 2.4 a), f induces a morphism U M(Hg). Proof of 0.3. By Proposition 1.4 the image of the monodromy representation of f lies in Sl 2 (R) G, for some compact group G, and its Zariski closure is Sl 2 (R) G. Hence, G Mon 0 (R) is again the product of Sl 2 (R) with a compact group. Lemma 2.4 c), implies that Hg(R) =Sl 2 (R) G for a compact group G, hence Hg(R)/K Sl 2 (R)/SO 2 is the upper half plane H. In particular, dim M(Hg) = 1. Since we assumed the family to be nonisotrivial and semistable, the morphism U M(Hg) is surjective. Consider the composition φ : U M(Hg) A g. Replacing U by an étale covering, we may assume that X 0 U is the pullback of a universal family of abelian varieties, defined over an étale covering A g of A g. The pull back of the tangent bundle on A g via φ is just φ T Ag = S 2 E 0,1 E 0,1 2. The differential dφ : T U φ T Ag E 0,1 2 is induced by the Kodaira- Spencer map E 1,0 T U E 0,1. By Proposition 1.4 E 1,0 E 0,1 =(L L 1 ) T, and the map dφ : T U E 0,1 2 lies in the component dφ : T U L 2 L 2 End(T).

21 A CHARACTERIZATION OF SHIMURA CURVES 253 This implies that the differential of the map U M(Hg) is no where vanishing, hence U M(Hg) is étale. q.e.d. Remarks 2.5. a) As is well-known (see [17], [18]) the moduli space of abelian varieties with a given special Mumford Tate group is necessarily a Satake holomorphic embedding. Hence the assumptions made in Proposition 0.3 imply in particular that the period map from U to the corresponding moduli space of abelian varieties with a fixed level structure is a Satake holomorphic embedding. b) Presumably Proposition 0.3 can also be obtained using [1]. Using Proposition 1.4 the maximality of the Higgs field should imply that the period map from U = H/Γ to the Siegel upper half plane is a rigid, totally geodesic, and equivariant holomorphic map. Then [1], Theorem 3.4, implies that f : X Y is a family of Mumford type, and as mentioned in the introduction one can finish the proof of Theorem 0.5 going through the classification of Shimura varieties. c) Without the assumption of rigidity, hidden behind the one saying that the maximal unitary local subsystem is defined over Q, we do not see a way to show directly, that the families are Kuga fibre spaces. One needs a precise description of the Z-structure on the decompositions of the variation of Hodge structures. On the other hand, the latter will allow one to prove Theorems 0.5 and 0.7 directly. d) Theorems 0.5 and 0.7 imply that all families f : X Y with maximal Higgs fields are Kuga fibre spaces, and that the period map is again a Satake holomorphic embedding. 3. Splitting over Q Up to now, we considered local systems of C-vector spaces induced by the family of abelian varieties. We say that a C-local system M is defined over a subring R of C, if there exists a local system M R of torsion-free R-modules with M = M R R C. In different terms, the representation γ M : π 0 (U, ) Gl(µ, C) is conjugate to one factoring like γ M : π 0 (U, ) Gl(µ, R) Gl(µ, C).

22 254 E. VIEHWEG & K. ZUO If M is defined over R, and if σ : R R is a ring isomorphism, we will write M σ R for the local system defined by γ M : π 0 (U, ) Gl(µ, R) σ Gl(µ, R ), and M σ = M σ R R C. In this section we want to show, that the splittings X = V U 1 and End(V) =W U considered in the last section are defined over Q, i.e., that there exists a number field K containing the field of definition for X and local K-subsystems with V K X K, U 1K X K, W K End(X K ) and U K End(X K ) X K = X L L K = V K U 1K, V K = W K U K, and with V = V K K C, U 1 = U 1K K C W = W K K C, U = U K K C. We start with a simple observation. Suppose that M is a local system defined over a number field L. The local system M L is given by a representation ρ : π 1 (U, ) Gl(M L ) for the fibre M L of M L over the base point. Fixing a positive integer r, let G(r, M) denote the set of all rank-r local subsystems of M and let Grass(r, M L ) be the Grassmann variety of r-dimensional subspaces. Then G(r, M) is the subvariety of Grass(r, M L ) Spec(L) Spec(C) consisting of the π 1 (U, )-invariant points. In particular, it is a projective variety defined over L. A K-valued point of G(r, M) corresponds to a local subsystem of M K = M L L K. One obtains the following well-known property: Lemma 3.1. If [W] G(r, M) is an isolated point, then W is defined over Q. In the proof of 3.7 we will also need: Lemma 3.2. Let M be a variation of Hodge structures defined over L, and let W M be an irreducible local subsystem of rank r defined over C,. Then W can be deformed to a local subsystem W t M, which is isomorphic to W and which is defined over a finite extension of L. Proof. By [4] M is completely reducible over C. decomposition M = W W. Hence we have a

23 A CHARACTERIZATION OF SHIMURA CURVES 255 The space G(r, M) of rank r local subsystems of M is defined over L and the subset {W t G(r, M); the composit W t W W pr 1 W is nonzero} forms a Zariski open subset. So there exists some W t in this subset, which is defined over some finite extension of L. Since p : W t W is nonzero, rank(w t ) = rank(w), and since W is irreducible, p is an isomorphism. q.e.d. Lemma 3.3. Let M be the underlying local system of a variation of Hodge structures of weight k defined over a number field L. Assume that there is a decomposition (3.3.1) M = U l i=1 M i in subvariations of Hodge structures, and let (3.3.2) (E,θ)=(N,0) l (F i,τ i = θ Fi ) be the induced decomposition of the Higgs field. Assume that width(m i ) = i, and that the M i have all generically maximal Higgs fields. Then the decomposition (3.3.1) is defined over Q. If L is real, it is defined over Q R. If M is polarized, then the decomposition (3.3.1) can be chosen to be orthogonal with respect to the polarization. Proof. Consider a family {W t } t of local subsystems of M defined over a disk with W 0 = M l.fort let (F Wt,τ t ) denote the Higgs bundle corresponding to W t. Hence (F Wt,τ t ) is obtained by restricting the F -filtration of M O U to W t O U and by taking the corresponding graded sheaf. So the Higgs map τ p,k p : F p,k p t i=1 F p 1,k p+1 t Ω 1 Y (log S) will again be generically isomorphic for t sufficiently closed to 0 and If the projection 2p k l and 2p 2 k l. ρ : W t M = U l l 1 M i U i=1 i=1 M i

24 256 E. VIEHWEG & K. ZUO is nonzero, the complete reducibility of local systems coming from variations of Hodge structures (see [4]) implies that W t contains an irreducible nontrivial direct factor, say W t which is isomorphic to a direct factor of U or of one of the local systems M i, for i<l. Restricting again the F filtration and passing to the corresponding graded sheaf, we obtain a Higgs bundle (F W t,τ t) with trivial Higgs field, or whose width is strictly smaller than l. On the other hand, (F W t,τ t) is a sub-higgs bundle of the Higgs bundle (F Wt,τ t ) of width l, a contradiction. So ρ is zero and W t = M l. Thus M l is rigid as a local subsystem of M, and by Lemma 3.1 M l is defined over Q. Assume now that L is real, hence M = M R C. The complex conjugation defines an involution ι on M. Let M l denote the image of M l under ι. Then M l has again generically isomorphic Higgs maps τ p,k p, for 2p k l and 2p 2 k l. If M l M l, repeating the argument used above, one obtains a map l 1 M l U M i, from a Higgs bundle of width l and with a maximal Higgs field to one with trivial Higgs field or of lower width. Again such a morphism must be zero, hence M l = M l in this case. So we can find a number field K, real in case L is real, and a local system M l,k M K with M l = M l,k K C. The polarization on M restricts to a nondegenerated intersection form on M K. Choosing for M l,k the orthogonal complement of M l,k in M K we obtain a splitting i=1 M K = M l,k M l,k inducing over C the splitting of the factor M l in (3.3.1). By induction on l we obtain 3.3. q.e.d. For a reductive algebraic group G and for a finitely generated group Γ let M(Γ,G) denote the moduli space of reductive representations of ΓinG. Theorem 3.4 (Simpson, [29], Corollary 9.18). Suppose Γ is a finitely generated group. Suppose φ : G H is a homomorphism of reductive

25 A CHARACTERIZATION OF SHIMURA CURVES 257 algebraic groups with finite kernel. Then the resulting morphism of moduli spaces φ : M(Γ,G) M(Γ,H) is finite. Corollary 3.5. Let Γ be π 1 (Y, ) of a projective manifold, and γ : Γ G be a reductive representation. If φγ M(Γ,H) comes from a C-variation of Hodge structures, then γ comes from a C-variation of Hodge structures as well. Proof. By Simpson a reductive local system is coming from a variation of Hodge structures if and only if the isomorphism class of the corresponding Higgs bundle is a fix point of the C -action. Since the C -action contains the identity and since it is compatible with φ, the finiteness of the preimage φ 1 φ(γ) implies that the isomorphism class of the Higgs bundle corresponding to γ is fixed by the C -action, as well. q.e.d. Definition 3.6. Let M be a local system of rank r, and defined over Q. Let γ M : π 1 (U, ) Sl(2, Q) be the corresponding representation of the fundamental group. For η π 1 (U, ) we write tr(γ M (η)) Q for the trace of η and tr(m) ={tr(γ M (η)); η π 1 (U, )}. Corollary 3.7. Under the assumptions made in 1.3: i. The splitting X = V U 1 is defined over Q, and over Q R in case L is real. If X is polarized, it can be chosen to be orthogonal. ii. The splitting End(V) = W U constructed in Lemma 1.7 is defined over Q, and over Q R in case L is real. If X is polarized, it can be chosen to be orthogonal. iii. Replacing Y by an étale covering Y, one can choose the decomposition V L T in 1.4 such that: a. L and T are defined over a number field K, real if L is real. b. One has an isomorphism V Q L Q Q T Q. c. tr(l) is a subset of the ring of integers O K of K. Proof. i) and ii) are direct consequences of 3.3. For iii) let us first remark that for L real, passing to an étale covering L and T can both be assumed to be defined over R. In fact, the local system L has a maximal Higgs field, hence its Higgs field is of the form (L L 1,τ ) where L is a theta characteristic. Hence it differs from L at most by the tensor

26 258 E. VIEHWEG & K. ZUO product with a two torsion point in Pic 0 (Y ). Replacing Y by an étale covering, we may assume L = L. From 1.4 d), we obtain T = T. Consider the isomorphism of local systems φ : L T V and the induced isomorphism φ 2 : End 0 (L T) =End 0 (L) End 0 (T) End 0 (L) End 0 (T) End(V). Since φ 2 End 0 (T) is the unitary part of this decomposition, by 3.3 it is defined over Q R, as well as φ 2 (End 0 (L) End 0 (T) End 0 (L)). The 1, 1 part of the Higgs field corresponding to φ 2 End 0 (L) has rank one, and its Higgs field is maximal. Hence φ 2 End 0 (L) is irreducible, and by 3.2 it is isomorphic to a local system, defined over Q. Hence T T End(T) and L L End(L) are both isomorphic to local systems defined over some real number field K.AnO K -structure can be defined by φ 2 (End(L)) OK = φ 2 (End(L)) K V OK. Consider for ν =2orν = g 0 the moduli space M(U, Sl(ν 2 )) of reductive representations of π(u, ) intosl(ν 2 ). It is a quasi-projective variety defined over Q. The fact that L L (or T T) is defined over Q implies that its isomorphy class in M(U, Sl(ν 2 )) is a Q valued point. Consider the morphism induced by the second tensor product ρ : M(U, Sl(ν)) M(U, Sl(ν 2 )) which is clearly defined over Q. By Theorem 3.4, ρ is finite, hence the fibre ρ 1 ([L L]) (or ρ 1 ([T T])) consists of finitely many Q-valued points, hence L and T can be defined over a number field K. If L is real, as already remarked above, we may choose K to be real. Obviously, for ρ π 1 (Y, ) one has tr(γ L (ρ)) 2 = tr(γ L L (ρ)). In fact, one may assume that γ L (ρ) is a diagonal matrix with entries a and b on the diagonal. Then tr(γ L L (ρ)) has a 2, b 2, ab and ba as diagonal elements. Since tr(γ L L (ρ)) O K we find tr(γ L (ρ)) O K. q.e.d. 4. Splitting over Q for S and isogenies In this section, we will consider the case L = Q and X Q = R 1 f Q X0, where f : X Y is a family of abelian varieties, S = Y \ U, and where the restriction X 0 U of f is a smooth family.

27 A CHARACTERIZATION OF SHIMURA CURVES 259 Lemma 4.1. Assume that S and let M Q be a Q-variation of Hodge structures of weight k and with unipotent monodromy around all points s S. Assume that over some number field K there exists a splitting M K = M Q Q K = W K U K where U = U K K C is unitary and where the Higgs field of W = W K K C is maximal. Then W, U and the decomposition M = W U are defined over Q. Moreover, U extends to a local system over Y. Proof. Let T be a local subsystem of W. Writing F p,q T, τ p,q, p+q=k p+q=k for the Higgs bundle corresponding to T, the maximality of the Higgs field for W implies that the Higgs field for T is maximal, as well. In particular, for all s S and for p>0 the residue maps res s (τ p,q ):F p,q T,s F p 1,q+1 T,s are isomorphisms. By [27] the residues of the Higgs field at s are defined by the nilpotent part of the local monodromy matrix around s. Hence if γ is a small loop around s in Y, and if ρ T (γ) denotes the image of γ under a representation of the fundamental group, defining T, the nilpotent part N(ρ T (γ)) = log ρ T (γ) ofρ T (γ) has to be nontrivial. We may assume that K is a Galois extension of Q. Recall that for σ Gal(K/Q) we denote the local systems obtained by composing the representation with σ by an upper index σ. Consider the composite p : U σ K M K = W K U K W K, and the induced map U σ = U σ K K C W. Let γ be a small loop around s S, and let ρ U (γ) and ρ U σ(γ) be the images of γ under the representations defining U and U σ respectively. Since U is unitary and unipotent, the nilpotent part of the monodromy matrix N(ρ U (γ)) = 0. This being invariant under conjugation, N(ρ U σ(γ)) is zero, as well as N(ρ p(u σ )(γ)). Therefore p(u σ ) = 0, hence U σ = U, and U is defined over Q. Taking again the orthogonal complement, one obtains the Q-splitting asked for in 4.1. Since N(ρ U (γ)) = 0, the residues of U are zero in all points s S, hence U extends to a local system on Y. q.e.d.

28 260 E. VIEHWEG & K. ZUO Corollary 4.2. Suppose that S. Then the splittings in Corollary 3.7 i) and ii), can be defined over Q. Lemma 4.3. Let M be a local system, defined over Z, and let M Q = W Q U Q be a decomposition, defined over Q. Then there exist local systems U Z and W Z, defined over Z with (4.3.1) U Q = U Z Q, W Q = W Z Q, and M Z W Z U Z. Moreover, if U Q is unitary with trivial local monodromies around S, then there exists an étale covering π : Y Y such that π U Q is trivial. Proof. Defining a Z-structure on W Q and U Q by W Z = W Q M Z and U Z = U Q M Z (4.3.1) obviously holds true. Since the integer elements of the unitary group form a finite group, the representation defining U factors through a finite quotient of the fundamental group π 1 (U, ) G. The condition on the local monodromies implies that this quotient factors through π 1 (Y, ), and we may choose Y to be the corresponding étale covering. q.e.d. By 4.2 we obtain decompositions R 1 f Q X0 = V Q U 1Q and End(V Q )=W Q U Q. By 4.1 the local monodromies of the unitary parts U 1 and U are trivial. Moreover, U is a subvariation of Hodge structures of weight 0, 0. Summing up, we obtain: Corollary 4.4. Let f : X Y be a family of abelian varieties with unipotent local monodromies around s S, and reaching the Arakelov bound. If S there exists a finite étale cover π : Y Y with: i. For a Z-variation of Hodge structures V Z of weight 1 with maximal Higgs field, we have π (R 1 f (Z X0 )) V Z Z2(g g 0), and π (R 1 f (Z X0 )) Q =(V Z Z2(g g 0) ) Q. ii. End(V Z ) W Z Zg2 0, End(V Z ) Q =(W Z Zg2 0 ) Q, where W Z is an Z-variation of Hodge structures of weight 0 with maximal Higgs field, and where Z g2 0 isalocalz-subsystem of type (0, 0).